seurat_pattern

1. Import Data and Library

library(Seurat)

## Attaching SeuratObject

library(SeuratData)

## Registered S3 method overwritten by 'cli':
##   method     from         
##   print.boxx spatstat.geom

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## ── Installed datasets ───────────────────────────────────── SeuratData v0.2.1 ──

## ✓ bmcite       0.3.0                    ✓ pbmcMultiome 0.1.0

## ────────────────────────────────────── Key ─────────────────────────────────────

## ✓ Dataset loaded successfully
## > Dataset built with a newer version of Seurat than installed
## ❓ Unknown version of Seurat installed

library(cowplot)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

load("seurat_data.RData")

# Import and Read Data
RNA_dat <- bm@assays$RNA
# RNA_mat <- t(as.matrix(RNA_dat@counts))

# Extract labels
cell_type <- bm$celltype.l2
cell_labels <- unique(cell_type)
sub_cluster_labels <- as.numeric(as.factor(sub_celltype))

load("seurat_corr.RData")

cor_pearson_mat <- cormat_list[[1]]; cor_spearman_mat <- cormat_list[[2]];
cor_kendall_mat <- cormat_list[[3]]; cor_hoeffd_mat <- cormat_list[[4]];
cor_blomqvist_mat <- cormat_list[[5]]; cor_dist_mat <- cormat_list[[6]];
cor_MI_mat <- cormat_list[[7]]; cor_XI_mat <- cormat_list[[8]];

2. Find contrast characteristics among the correlation coefficients above

2-1-1. Low Pearson (< 0.40) and High Spearman (> 0.60) (linearity vs monotone)

cor_contrast1 <- (abs(cor_pearson_mat) < 0.4) & (abs(cor_spearman_mat) > 0.6)
cor_contrast_ind1 <- which(cor_contrast1, arr.ind = T)
nrow(cor_contrast_ind1)

## [1] 3

2-1-2. Visualization of Low Pearson (< 0.40) and High Spearman (> 0.60) (linearity vs monotone)

par(mfrow = c(2, 2))
for (i in 1:nrow(cor_contrast_ind1)){
  index1 <- cor_contrast_ind1[i, 1]; index2 <- cor_contrast_ind1[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3))))
}

2-2-1. High Pearson (> 0.85) and Low Spearman (< 0.15) (linearity vs monotone)

cor_contrast2 <- (abs(cor_pearson_mat) > 0.85) & (abs(cor_spearman_mat) < 0.15)
cor_contrast_ind2 <- which(cor_contrast2, arr.ind = T)
nrow(cor_contrast_ind2)

## [1] 4

2-2-2. Visualization of High Pearson (> 0.85) and Low Spearman (< 0.15) (linearity vs monotone)

par(mfrow = c(2, 2))
for (i in 1:nrow(cor_contrast_ind2)){
  index1 <- cor_contrast_ind2[i, 1]; index2 <- cor_contrast_ind2[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3))))
}

2-2-1. Low Pearson (< 0.50) and High Kendall (> 0.50) (linearity vs monotone)

cor_contrast3 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_kendall_mat) > 0.5)
cor_contrast_ind3 <- which(cor_contrast3, arr.ind = T)
nrow(cor_contrast_ind3)

## [1] 14

2-2-2. Visualization of Low Pearson (< 0.50) and High Kendall (> 0.50) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind3[i, 1]; index2 <- cor_contrast_ind3[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3))))
}

2-4-1. High Pearson (> 0.85) and Low Kendall (< 0.15) (linearity vs monotone)

cor_contrast4 <- (abs(cor_pearson_mat) > 0.85) & (abs(cor_kendall_mat) < 0.15)
cor_contrast_ind4 <- which(cor_contrast4, arr.ind = T)
nrow(cor_contrast_ind4)

## [1] 7

2-4-2. Visualization of High Pearson (> 0.85) and Low Kendall (< 0.15) (linearity vs monotone)

par(mfrow = c(2, 4))
for (i in 1:nrow(cor_contrast_ind4)){
  index1 <- cor_contrast_ind4[i, 1]; index2 <- cor_contrast_ind4[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3))))
}

2-5-1. Low Pearson (< 0.50) and High Hoeffding’s D (> 0.50) (linearity vs monotone)

cor_contrast5 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_hoeffd_mat) > 0.5)
cor_contrast_ind5 <- which(cor_contrast5, arr.ind = T)
nrow(cor_contrast_ind5)

## [1] 0

2-6-1. High Pearson (> 0.90) and Low Hoeffding’s D (< 0.05) (linearity vs monotone)

cor_contrast6 <- (abs(cor_pearson_mat) > 0.9) & (abs(cor_hoeffd_mat) < 0.05)
cor_contrast_ind6 <- which(cor_contrast6, arr.ind = T)
nrow(cor_contrast_ind6)

## [1] 9

2-6-2. Visualization of High Pearson (> 0.90) and Low Hoeffding’s D (< 0.05) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind6)){
  index1 <- cor_contrast_ind6[i, 1]; index2 <- cor_contrast_ind6[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Hoeffding's D of ", round(cor_hoeffd_mat[index1, index2], 3))))
}

2-7-1. Low Pearson (< 0.05) and high Blomqvist’s beta (> 0.90) (linearity vs monotone)

cor_contrast7 <- (abs(cor_pearson_mat) < 0.05) & (abs(cor_blomqvist_mat) > 0.9)
cor_contrast_ind7 <- which(cor_contrast7, arr.ind = T)
nrow(cor_contrast_ind7)

## [1] 7

2-7-2. Visualization of Low Pearson (< 0.05) and high Blomqvist’s beta (> 0.90) (linearity vs monotone)

par(mfrow = c(2, 4))
for (i in 1:nrow(cor_contrast_ind7)){
  index1 <- cor_contrast_ind7[i, 1]; index2 <- cor_contrast_ind7[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

2-8-1. High Pearson (> 0.90) and Low Blomqvist’s beta (< 0.20) (linearity vs monotone)

cor_contrast8 <- (abs(cor_pearson_mat) > 0.9) & (abs(cor_blomqvist_mat) < 0.2)
cor_contrast_ind8 <- which(cor_contrast8, arr.ind = T)
nrow(cor_contrast_ind8)

## [1] 6

2-8-2. Visualization of High Pearson (> 0.90) and Low Blomqvist’s beta (< 0.20) (linearity vs monotone)

par(mfrow = c(2, 3))
for (i in 1:nrow(cor_contrast_ind8)){
  index1 <- cor_contrast_ind8[i, 1]; index2 <- cor_contrast_ind8[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

2-9-1. Low Pearson (< 0.50) and High XI (> 0.50) (linearity vs non-linearity)

cor_contrast9 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind9 <- which(cor_contrast9, arr.ind = T)
nrow(cor_contrast_ind9)

## [1] 0

2-10-1. High Pearson (> 0.90) and Low XI (< 0.15) (linearity vs non-linearity)

cor_contrast10 <- (abs(cor_pearson_mat) > 0.9) & (abs(cor_XI_mat) < 0.15)
cor_contrast_ind10 <- which(cor_contrast10, arr.ind = T)
nrow(cor_contrast_ind10)

## [1] 8

2-10-2. Visualization of High Pearson (> 0.90) and Low XI (< 0.15) (linearity vs non-linearity)

par(mfrow = c(2, 4))
for (i in 1:nrow(cor_contrast_ind10)){
  index1 <- cor_contrast_ind10[i, 1]; index2 <- cor_contrast_ind10[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

2-11-1. Low Spearman (< 0.05) and High Distance Correlation (> 0.95) (monotone vs non-linearity)

cor_contrast11 <- (abs(cor_spearman_mat) < 0.05) & (abs(cor_dist_mat) > 0.95)
cor_contrast_ind11 <- which(cor_contrast11, arr.ind = T)
nrow(cor_contrast_ind11)

## [1] 10

2-11-2. Visualization of Low Spearman (< 0.05) and High Distance Correlation (> 0.95) (monotone vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind11)){
  index1 <- cor_contrast_ind11[i, 1]; index2 <- cor_contrast_ind11[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

2-12-1. High Spearman (> 0.60) and Low Distance Correlation (< 0.35) (monotone vs non-linearity)

cor_contrast12 <- (abs(cor_spearman_mat) > 0.6) & (abs(cor_dist_mat) < 0.35)
cor_contrast_ind12 <- which(cor_contrast12, arr.ind = T)
nrow(cor_contrast_ind12)

## [1] 6

2-12-2. Visualization of High Spearman (> 0.60) and Low Distance Correlation (< 0.35) (monotone vs non-linearity)

par(mfrow = c(2, 3))
for (i in 1:nrow(cor_contrast_ind12)){
  index1 <- cor_contrast_ind12[i, 1]; index2 <- cor_contrast_ind12[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

2-13-1. Low Kendall (< 0.03) and High Distance Correlation (> 0.95) (monotone vs non-linearity)

cor_contrast13 <- (abs(cor_kendall_mat) < 0.03) & (abs(cor_dist_mat) > 0.95)
cor_contrast_ind13 <- which(cor_contrast13, arr.ind = T)
nrow(cor_contrast_ind13)

## [1] 5

2-13-2. Visualization of Low Kendall (< 0.03) and High Distance Correlation (> 0.95) (monotone vs non-linearity)

par(mfrow = c(2, 3))
for (i in 1:nrow(cor_contrast_ind13)){
  index1 <- cor_contrast_ind13[i, 1]; index2 <- cor_contrast_ind13[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

2-14-1. High Kendall (> 0.50) and Low Distance Correlation (< 0.50) (monotone vs non-linearity)

cor_contrast14 <- (abs(cor_kendall_mat) > 0.5) & (abs(cor_dist_mat) < 0.5)
cor_contrast_ind14 <- which(cor_contrast14, arr.ind = T)
nrow(cor_contrast_ind14)

## [1] 46

2-14-2. Visualization of High Kendall (> 0.50) and Low Distance Correlation (< 0.50) (monotone vs non-linearity)

par(mfrow = c(2, 3))
for (i in 1:10){
  index1 <- cor_contrast_ind14[i, 1]; index2 <- cor_contrast_ind14[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

2-15-1. Low Distance Correlation (< 0.50) and High Hoeffiding’s D (> 0.50) (non-linearity vs monotone)

cor_contrast15 <- (abs(cor_dist_mat) < 0.5) & (abs(cor_hoeffd_mat) > 0.5)
cor_contrast_ind15 <- which(cor_contrast15, arr.ind = T)
nrow(cor_contrast_ind15)

## [1] 0

2-16-1. High Distance Correlation (> 0.90) and Low Hoeffiding’s D (< 0.10) (non-linearity vs monotone)

cor_contrast16 <- (abs(cor_dist_mat) > 0.90) & (abs(cor_hoeffd_mat) < 0.10)
cor_contrast_ind16 <- which(cor_contrast16, arr.ind = T)
nrow(cor_contrast_ind16)

## [1] 1347

2-16-2. Visualization of High Distance Correlation (> 0.90) and Low Hoeffiding’s D (< 0.10) (non-linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind16[i, 1]; index2 <- cor_contrast_ind16[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("Hoeffiding's D of ", round(cor_hoeffd_mat[index1, index2], 3))))
}

2-17-1. Low Distance Correlation (< 0.50) and High XI (> 0.50) (non-linearity vs non-linearity)

cor_contrast17 <- (abs(cor_dist_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind17 <- which(cor_contrast17, arr.ind = T)
nrow(cor_contrast_ind17)

## [1] 0

2-18-1. High Distance Correlation (> 0.90) and Low XI (< 0.10) (non-linearity vs non-linearity)

cor_contrast18 <- (abs(cor_dist_mat) > 0.9) & (abs(cor_XI_mat) < 0.1)
cor_contrast_ind18 <- which(cor_contrast18, arr.ind = T)
nrow(cor_contrast_ind18)

## [1] 651

2-18-2. Visualization of High Distance Correlation (> 0.90) and Low XI (< 0.10) (non-linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind18[i, 1]; index2 <- cor_contrast_ind18[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

2-19-1. Low Distance Correlation (< 0.10) and High Blomqvist’s Beta (> 0.90) (non-linearity vs non-linearity)

cor_contrast19 <- (abs(cor_dist_mat) < 0.1) & (abs(cor_blomqvist_mat) > 0.9)
cor_contrast_ind19 <- which(cor_contrast19, arr.ind = T)
nrow(cor_contrast_ind19)

## [1] 21

2-19-2. Visualization of Low Distance Correlation (< 0.10) and High Blomqvist’s Beta (> 0.90) (non-linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind19[i, 1]; index2 <- cor_contrast_ind19[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("Blomqvist's Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

2-20-1. High Distance Correlation (> 0.90) and Low Blomqvist’s Beta (< 0.10) (non-linearity vs non-linearity)

cor_contrast20 <- (abs(cor_dist_mat) > 0.9) & (abs(cor_blomqvist_mat) < 0.1)
cor_contrast_ind20 <- which(cor_contrast20, arr.ind = T)
nrow(cor_contrast_ind20)

## [1] 286

2-20-2. Visualization of High Distance Correlation (> 0.90) and Low Blomqvist’s Beta (< 0.10) (non-linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind20[i, 1]; index2 <- cor_contrast_ind20[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("Blomqvist's Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

3. Explore dense plots

3-1. Visualize dense plots

dense_ind <- t(matrix(c(1, 210, 1, 333, 2, 415, 3, 4, 3, 5, 4, 5,
                        15, 47, 15, 101, 15, 173, 15, 358),
                      nrow = 2))
cluster_ind <- dense_ind[c(1,2,4,7,8,9,10),]

par(mfrow = c(2, 3))
for (i in 1:nrow(dense_ind)){
  index1 <- dense_ind[i, 1]; index2 <- dense_ind[i, 2];
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"))
}

3-2. Visualize with alpha parameter (transparency)

par(mfrow = c(2, 3))
for (i in 1:nrow(dense_ind)){
  index1 <- dense_ind[i, 1]; index2 <- dense_ind[i, 2];
  plot(sub_dat[,index1], sub_dat[,index2], col = scales::alpha(sub_cluster_labels, 0.2), asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"))
}

3-3. Visualize with contour plots (density)

#' Plotting density
#'
#' @param mat matrix
#' @param grid_spacing numeric vector
#' @param n integer for \code{stats::kde}
#' @param max_num integer
#' @param density_colors color vector
#' @param point_color color
#' @param contour_color color
#' @param grid_color color
#' @param pch integer
#' @param contour_lwd numeric
#' @param grid_lty integer
#' @param grid_lwd numeric
#' @param ... additional graphical parameters
#'
#' @return nothing
#' @export
plot_density <- function(mat, grid_spacing,
                         xlim,
                         ylim,
                         n = 100,
                         max_num = 20000,
                         density_colors = grDevices::heat.colors(100, alpha = 0.8),
                         point_color = grDevices::rgb(0.5, 0.5, 0.5, 0.8),
                         contour_color = grDevices::rgb(1,1,1,0.5),
                         grid_color = "gray",
                         xaxt = "n",
                         yaxt = "n",
                         axes = F,
                         xlab = "",
                         ylab = "",
                         pch = 16,
                         contour_lwd = 2,
                         grid_lty = 2,
                         grid_lwd = 0.5,
                         ...){
  stopifnot(ncol(mat) == 2)

  mat2 <- rbind(mat,
                c(xlim[1], ylim[1]),
                c(xlim[1], ylim[2]),
                c(xlim[2], ylim[1]),
                c(xlim[2], ylim[2]))
  f1 <- MASS::kde2d(mat2[,1], mat2[,2], n = 100)
  graphics::image(f1,
                  col = density_colors,
                  xlim = xlim,
                  ylim = ylim,
                  ...)

  for(i in grid_spacing){
    graphics::lines(rep(i,2), c(-2,2)*max(abs(grid_spacing)),
                    lty = grid_lty,
                    lwd = grid_lwd,
                    col = grid_color)
    graphics::lines(c(-2,2)*max(abs(grid_spacing)), rep(i,2),
                    lty = grid_lty,
                    lwd = grid_lwd,
                    col = grid_color)
  }

  if(nrow(mat) > max_num) {
    idx <- sample(1:nrow(mat), size = max_num)
  } else {
    idx <- 1:nrow(mat)
  }
  graphics::points(mat[idx,1], mat[idx,2],
                   col = point_color,
                   pch = pch)

  graphics::contour(f1, add = T,
                    drawlabels = F,
                    col = contour_color,
                    lwd = contour_lwd)

  invisible()
}

par(mfrow = c(1, 2))
for (i in 1:nrow(dense_ind)){
  index1 <- dense_ind[i, 1]; index2 <- dense_ind[i, 2];
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"))
  xylim <- par("usr")
  
  x <- sub_dat[,index1]; y <- sub_dat[,index2];
  xmin <- min(x); xmax <- max(x); ymin <- min(y); ymax <- max(y);
  total_min <- min(xmin, ymin); total_max <- max(xmax, ymax);
  
  plot_density(cbind(x, y),
               grid_spacing = seq(total_min, total_max, by=((total_max - total_min) / 10)),
               xlim = c(xylim[1], xylim[2]), ylim = c(xylim[3], xylim[4]),
               xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
               ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"))
  
}

par(mfrow = c(1,2))
for (i in 1:nrow(cluster_ind)){
  index1 <- cluster_ind[i, 1]; index2 <- cluster_ind[i, 2];
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"))
  
  x <- sub_dat[,index1]; y <- sub_dat[,index2];
  xmin <- min(x); xmax <- max(x); ymin <- min(y); ymax <- max(y);
  total_min <- min(xmin, ymin); total_max <- max(xmax, ymax);
  
  plot_density(cbind(x, y),
               grid_spacing = seq(total_min, total_max, by = 10),
               xlim = c(-20, 60), ylim = c(-50, 50),
               xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
               ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"))
}

4. Define a function to exclude repeated indices

contrast_helper <- function(contrast_ind, threshold){
  to_track <- data.frame()
  
  for (i in 1:nrow(contrast_ind)){
    ind1 <- contrast_ind[i, 1]; ind2 <- contrast_ind[i, 2];
    
    if (i == 1){ # first element
      to_track <- data.frame(row.names = c(ind1, ind2), val = c(1, 1))
      ans <- matrix(contrast_ind[i, ], nrow = 1)
    } else { 
      if (nrow(ans) == 10) { # want to extract 10 pairs
        break
      } else {
        row_names <- row.names(to_track)
        ind1_str <- toString(ind1); ind2_str <- toString(ind2); 
        
        if (ind1_str %in% row_names){
          first_bool <- to_track[ind1_str, ] < threshold
          to_track[ind1_str, ] <- to_track[ind1_str, ] + 1
        } else {
          to_track[ind1_str, ] <- 1
          first_bool <- TRUE
        }
        
        if (ind2_str %in% row_names){
          second_bool <- to_track[ind2_str, ] < threshold
          to_track[ind2_str, ] <- to_track[ind2_str, ] + 1
        } else {
          to_track[ind2_str, ] <- 1
          second_bool <- TRUE
        }
        
        if (first_bool & second_bool){
          ans <- rbind(ans, contrast_ind[i,])
        }
      }
    }
  }
  return (ans)
}

4-1-1. Low Pearson (< 0.40) and High Spearman (> 0.60) (linearity vs monotone)

cor_contrast1 <- (abs(cor_pearson_mat) < 0.4) & (abs(cor_spearman_mat) > 0.6)
cor_contrast_ind1 <- which(cor_contrast1, arr.ind = T)
cor_contrast_ind1 <- contrast_helper(cor_contrast_ind1, 3)
nrow(cor_contrast_ind1)

## [1] 3

4-1-2. Visualization of Low Pearson (< 0.40) and High Spearman (> 0.60) (linearity vs monotone)

par(mfrow = c(2, 2))
for (i in 1:nrow(cor_contrast_ind1)){
  index1 <- cor_contrast_ind1[i, 1]; index2 <- cor_contrast_ind1[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3))))
}

4-2-1. High Pearson (> 0.89) and Low Spearman (< 0.15) (linearity vs monotone)

cor_contrast2 <- (abs(cor_pearson_mat) > 0.80) & (abs(cor_spearman_mat) < 0.15)
cor_contrast_ind2 <- which(cor_contrast2, arr.ind = T)
cor_contrast_ind2 <- contrast_helper(cor_contrast_ind2, 3)

nrow(cor_contrast_ind2)

## [1] 6

4-2-2. Visualization of High Pearson (> 0.89) and Low Spearman (< 0.15) (linearity vs monotone)

par(mfrow = c(2, 3))
for (i in 1:nrow(cor_contrast_ind2)){
  index1 <- cor_contrast_ind2[i, 1]; index2 <- cor_contrast_ind2[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3))))
}

4-4-1. Low Pearson (< 0.50) and High Kendall (> 0.50) (linearity vs monotone)

cor_contrast3 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_kendall_mat) > 0.5)
cor_contrast_ind3 <- which(cor_contrast3, arr.ind = T)
cor_contrast_ind3 <- contrast_helper(cor_contrast_ind3, 3)
nrow(cor_contrast_ind3)

## [1] 8

4-4-2. Visualization of Low Pearson (< 0.50) and High Kendall (> 0.50) (linearity vs monotone)

par(mfrow = c(2, 4))
for (i in 1:nrow(cor_contrast_ind3)){
  index1 <- cor_contrast_ind3[i, 1]; index2 <- cor_contrast_ind3[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3))))
}

4-4-1. High Pearson (> 0.80) and Low Kendall (< 0.20) (linearity vs monotone)

cor_contrast4 <- (abs(cor_pearson_mat) > 0.80) & (abs(cor_kendall_mat) < 0.20)
cor_contrast_ind4 <- which(cor_contrast4, arr.ind = T)
cor_contrast_ind4 <- contrast_helper(cor_contrast_ind4, 3)
nrow(cor_contrast_ind4)

## [1] 10

4-4-2. Visualization of High Pearson (> 0.80) and Low Kendall (< 0.20) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind4)){
  index1 <- cor_contrast_ind4[i, 1]; index2 <- cor_contrast_ind4[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3))))
}

4-5-1. Low Pearson (< 0.50) and High Hoeffding’s D (> 0.50) (linearity vs monotone)

cor_contrast5 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_hoeffd_mat) > 0.5)
cor_contrast_ind5 <- which(cor_contrast5, arr.ind = T)
nrow(cor_contrast_ind5)

## [1] 0

4-6-1. High Pearson (> 0.90) and Low Hoeffding’s D (< 0.1) (linearity vs monotone)

cor_contrast6 <- (abs(cor_pearson_mat) > 0.9) & (abs(cor_hoeffd_mat) < 0.1)
cor_contrast_ind6 <- which(cor_contrast6, arr.ind = T)
cor_contrast_ind6 <- contrast_helper(cor_contrast_ind6, 3)
nrow(cor_contrast_ind6)

## [1] 10

4-6-2. Visualization of High Pearson (> 0.90) and Low Hoeffding’s D (< 0.1) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind6)){
  index1 <- cor_contrast_ind6[i, 1]; index2 <- cor_contrast_ind6[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Hoeffding's D of ", round(cor_hoeffd_mat[index1, index2], 3))))
}

4-7-1. Low Pearson (< 0.15) and high Blomqvist’s beta (> 0.9) (linearity vs monotone)

cor_contrast7 <- (abs(cor_pearson_mat) < 0.15) & (abs(cor_blomqvist_mat) > 0.9)
cor_contrast_ind7 <- which(cor_contrast7, arr.ind = T)
cor_contrast_ind7 <- contrast_helper(cor_contrast_ind7, 3)
nrow(cor_contrast_ind7)

## [1] 10

4-7-2. Visualization of Low Pearson (< 0.15) and high Blomqvist’s beta (> 0.9) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind7)){
  index1 <- cor_contrast_ind7[i, 1]; index2 <- cor_contrast_ind7[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

4-8-1. High Pearson (> 0.85) and Low Blomqvist’s beta (< 0.15) (linearity vs monotone)

cor_contrast8 <- (abs(cor_pearson_mat) > 0.85) & (abs(cor_blomqvist_mat) < 0.15)
cor_contrast_ind8 <- which(cor_contrast8, arr.ind = T)
cor_contrast_ind8 <- contrast_helper(cor_contrast_ind8, 3)
nrow(cor_contrast_ind8)

## [1] 10

4-8-2. Visualization of High Pearson (> 0.85) and Low Blomqvist’s beta (< 0.15) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind8)){
  index1 <- cor_contrast_ind8[i, 1]; index2 <- cor_contrast_ind8[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

4-9-1. Low Pearson (< 0.50) and High XI (> 0.50) (linearity vs non-linearity)

cor_contrast9 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind9 <- which(cor_contrast9, arr.ind = T)
nrow(cor_contrast_ind9)

## [1] 0

4-10-1. High Pearson (> 0.85) and Low XI (< 0.15) (linearity vs non-linearity)

cor_contrast10 <- (abs(cor_pearson_mat) > 0.85) & (abs(cor_XI_mat) < 0.15)
cor_contrast_ind10 <- which(cor_contrast10, arr.ind = T)
cor_contrast_ind10 <- contrast_helper(cor_contrast_ind10, 3)
nrow(cor_contrast_ind10)

## [1] 10

4-10-2. Visualization of High Pearson (> 0.85) and Low XI (< 0.15) (linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind10)){
  index1 <- cor_contrast_ind10[i, 1]; index2 <- cor_contrast_ind10[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                    "\n",
                    paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

4-11-1. Low Spearman (< 0.10) and High Distance Correlation (> 0.90) (monotone vs non-linearity)

cor_contrast11 <- (abs(cor_spearman_mat) < 0.10) & (abs(cor_dist_mat) > 0.90)
cor_contrast_ind11 <- which(cor_contrast11, arr.ind = T)
cor_contrast_ind11 <- contrast_helper(cor_contrast_ind11, 3)
nrow(cor_contrast_ind11)

## [1] 10

4-11-2. Visualization of Low Spearman (< 0.10) and High Distance Correlation (> 0.90) (monotone vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind11)){
  index1 <- cor_contrast_ind11[i, 1]; index2 <- cor_contrast_ind11[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

4-12-1. High Spearman (> 0.60) and Low Distance Correlation (< 0.4) (monotone vs non-linearity)

cor_contrast12 <- (abs(cor_spearman_mat) > 0.6) & (abs(cor_dist_mat) < 0.4)
cor_contrast_ind12 <- which(cor_contrast12, arr.ind = T)
cor_contrast_ind12 <- contrast_helper(cor_contrast_ind12, 3)
nrow(cor_contrast_ind12)

## [1] 9

4-12-2. Visualization of High Spearman (> 0.60) and Low Distance Correlation (< 0.4) (monotone vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind12)){
  index1 <- cor_contrast_ind12[i, 1]; index2 <- cor_contrast_ind12[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

4-13-1. Low Kendall (< 0.10) and High Distance Correlation (> 0.90) (monotone vs non-linearity)

cor_contrast13 <- (abs(cor_kendall_mat) < 0.10) & (abs(cor_dist_mat) > 0.90)
cor_contrast_ind13 <- which(cor_contrast13, arr.ind = T)
cor_contrast_ind13 <- contrast_helper(cor_contrast_ind13, 3)
nrow(cor_contrast_ind13)

## [1] 10

4-13-2. Visualization of Low Kendall (< 0.10) and High Distance Correlation (> 0.90) (monotone vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind13)){
  index1 <- cor_contrast_ind13[i, 1]; index2 <- cor_contrast_ind13[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

4-14-1. High Spearman (> 0.60) and Low Distance Correlation (< 0.4) (monotone vs non-linearity)

cor_contrast14 <- (abs(cor_spearman_mat) > 0.6) & (abs(cor_dist_mat) < 0.4)
cor_contrast_ind14 <- which(cor_contrast14, arr.ind = T)
cor_contrast_ind14 <- contrast_helper(cor_contrast_ind14, 3)
nrow(cor_contrast_ind14)

## [1] 9

4-14-2. Visualization of High Spearman (> 0.60) and Low Distance Correlation (< 0.4) (monotone vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind14)){
  index1 <- cor_contrast_ind14[i, 1]; index2 <- cor_contrast_ind14[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

4-15-1. Low Kendall (< 0.05) and High Distance Correlation (> 0.90) (monotone vs non-linearity)

cor_contrast15 <- (abs(cor_kendall_mat) < 0.05) & (abs(cor_dist_mat) > 0.90)
cor_contrast_ind15 <- which(cor_contrast15, arr.ind = T)
cor_contrast_ind15 <- contrast_helper(cor_contrast_ind15, 3)
nrow(cor_contrast_ind15)

## [1] 6

4-15-2. Visualization of Low Kendall (< 0.05) and High Distance Correlation (> 0.90) (monotone vs non-linearity)

par(mfrow = c(2, 3))
for (i in 1:nrow(cor_contrast_ind15)){
  index1 <- cor_contrast_ind15[i, 1]; index2 <- cor_contrast_ind15[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

4-16-1. High Kendall (> 0.50) and Low Distance Correlation (< 0.50) (monotone vs non-linearity)

cor_contrast16 <- (abs(cor_kendall_mat) > 0.5) & (abs(cor_dist_mat) < 0.5)
cor_contrast_ind16 <- which(cor_contrast16, arr.ind = T)
cor_contrast_ind16 <- contrast_helper(cor_contrast_ind16, 3)
nrow(cor_contrast_ind16)

## [1] 10

4-16-2. Visualization of High Kendall (> 0.50) and Low Distance Correlation (< 0.50) (monotone vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind16)){
  index1 <- cor_contrast_ind16[i, 1]; index2 <- cor_contrast_ind16[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                    "\n",
                    paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

4-17-1. Low Distance Correlation (< 0.50) and High Hoeffiding’s D (> 0.50) (non-linearity vs monotone)

cor_contrast17 <- (abs(cor_dist_mat) < 0.5) & (abs(cor_hoeffd_mat) > 0.5)
cor_contrast_ind17 <- which(cor_contrast17, arr.ind = T)
nrow(cor_contrast_ind17)

## [1] 0

4-18-1. High Distance Correlation (> 0.90) and Low Hoeffiding’s D (< 0.10) (non-linearity vs monotone)

cor_contrast18 <- (abs(cor_dist_mat) > 0.90) & (abs(cor_hoeffd_mat) < 0.10)
cor_contrast_ind18 <- which(cor_contrast18, arr.ind = T)
cor_contrast_ind18 <- contrast_helper(cor_contrast_ind18, 3)

nrow(cor_contrast_ind18)

## [1] 10

4-18-2. Visualization of High Distance Correlation (> 0.90) and Low Hoeffiding’s D (< 0.10) (non-linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind18[i, 1]; index2 <- cor_contrast_ind18[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 18, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("Hoeffiding's D of ", round(cor_hoeffd_mat[index1, index2], 3))))
}

4-19-1. Low Distance Correlation (< 0.50) and High XI (> 0.50) (non-linearity vs non-linearity)

cor_contrast19 <- (abs(cor_dist_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind19 <- which(cor_contrast19, arr.ind = T)
nrow(cor_contrast_ind19)

## [1] 0

4-20-1. High Distance Correlation (> 0.90) and Low XI (< 0.10) (non-linearity vs non-linearity)

cor_contrast20 <- (abs(cor_dist_mat) > 0.9) & (abs(cor_XI_mat) < 0.1)
cor_contrast_ind20 <- which(cor_contrast20, arr.ind = T)
cor_contrast_ind20 <- contrast_helper(cor_contrast_ind20, 3)

nrow(cor_contrast_ind20)

## [1] 10

4-20-2. Visualization of High Distance Correlation (> 0.90) and Low XI (< 0.10) (non-linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind20[i, 1]; index2 <- cor_contrast_ind20[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

4-21-1. Low Distance Correlation (< 0.10) and High Blomqvist’s Beta (> 0.90) (non-linearity vs non-linearity)

cor_contrast21 <- (abs(cor_dist_mat) < 0.1) & (abs(cor_blomqvist_mat) > 0.9)
cor_contrast_ind21 <- which(cor_contrast21, arr.ind = T)
cor_contrast_ind21 <- contrast_helper(cor_contrast_ind21, 3)
nrow(cor_contrast_ind21)

## [1] 10

4-21-2. Visualization of Low Distance Correlation (< 0.10) and High Blomqvist’s Beta (> 0.90) (non-linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind21[i, 1]; index2 <- cor_contrast_ind21[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("Blomqvist's Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

4-22-1. High Distance Correlation (> 0.90) and Low Blomqvist’s Beta (< 0.10) (non-linearity vs non-linearity)

cor_contrast22 <- (abs(cor_dist_mat) > 0.9) & (abs(cor_blomqvist_mat) < 0.1)
cor_contrast_ind22 <- which(cor_contrast22, arr.ind = T)
cor_contrast_ind22 <- contrast_helper(cor_contrast_ind22, 3)
nrow(cor_contrast_ind22)

## [1] 10

4-22-2. Visualization of High Distance Correlation (> 0.90) and Low Blomqvist’s Beta (< 0.10) (non-linearity vs non-linearity)

par(mfrow = c(2, 5))
for (i in 1:10){
  index1 <- cor_contrast_ind22[i, 1]; index2 <- cor_contrast_ind22[i, 2]
  plot(sub_dat[,index1], sub_dat[,index2], col = sub_cluster_labels, asp = T,
       pch = 16, xlab = paste0(colnames(sub_dat)[index1], ", (", index1, ")"),
       ylab = paste0(colnames(sub_dat)[index2], ", (", index2, ")"), 
       main = paste(paste0("Dist. Cor of ", round(cor_dist_mat[index1, index2], 3)),
                    "\n",
                    paste0("Blomqvist's Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}